# ZZ-reduction

ZZ-reduction, also called ZZ-r, is a 2LLL variant of the ZZ method for 3x3x3 proposed by Adrian Currier in 2014.

During insertion of last slot of F2L, two or all four opposite edges of the last layer are permuted. Next, a set of OLL algorithms that preserve edge permutation are used. Finally, a subset of PLL solves the cube. ZZ-r has the lowest algorithm count of any 2LLL method, needing only 16 algorithms (14 excluding mirrors). If superphasing is used during last slot of F2L, whereby all last layer edges are permuted every time, the algorithm count is even less at 12 (10 excluding mirrors).

You can use ZZ-r as a transition to full ZZ or CFOP. Full ZZ or CFOP might be faster than ZZ-r once you memorize all 28 or 78 algorithms, respectively.

## The Steps

### EOLine + F2L

See EOLine

This part is the same as the traditional ZZ method.

### Phasing

After EOLine is completed, Phasing is employed during the insertion of the final block pair in F2L, with LL edges resulting in opposite colours (blue/green and orange/red) being placed opposite each other. This leaves edges either in parity (opposite but not permuted) or solved (permuted).

However, you can also employ superphasing, whereby you always permute all edges of the last layer. One way of achieving this is by first phasing, then if you notice that you have parity employing this algorithm:

R' U' R U' R' U2 R2 then phase (U R' U R U2 R')

### OLC

See OLC Algorithms

OLC (Orient Last Corners) is a subset of OLL that orients corners only and also preserves two opposite or all four edge permutations.

### PLL

See PLL Algorithms

Since the phased edges were preserved in OLC, you should end up with only 9 possible PLL cases, down from 21 in full PLL. These are Aa/b, E, F, H, Na/b, T, Z.

But, if you intentionally phase to permute all four edges (superphase) every time and then employ OLCs that preserve this edge permutation, you can further limit PLL to just 4 cases (Aa/b, E, and H).

One other option is to limit PLL to a different set of 16 cases (Aa/b, E, Ga/b/c/d, H, Ja/b, Ra/b, Ua/b, V, Y) by antiphasing. That is, instead of placing LL opposite edge colors when completing F2L, placing adjacent edge colors. As a suggestion, this could be used to finish memorizing the remaining PLL cases of full ZZ or CFOP after memorizing the smaller PLL subset that results from phasing or superphasing.

## PLL probabilities

The probabilities for the PLL subset in ZZ-r are different than for full PLL. The probability of skipping PLL is 1/24, while the probabilities of getting each case are as follows:

• A 1/3
• F 1/6
• T 1/6
• E 1/12
• N 1/12
• Z 1/12
• H 1/24

If superphasing and OLCs that preserve all edge permutations are employed, the probability of skipping PLL is 1/12, while the probabilities of getting each case are as follows:

• A 2/3
• E 1/6
• H 1/12

## Pros

• Reduced algorithm count: Only 16 algorithms are needed for full ZZ-r (14 excluding mirror cases). This is a lower alg count than any other 2LLL method. If using superphasing, only 12 algorithms are needed (10 excluding mirror cases). Technically, superphasing makes ZZ-r a 3LLL method.
• Faster recognition: Because there are so few last layer cases, recognition is very quick.
• Frequent skips: The low number of last layer cases make for more frequent skip cases. Skip probabilities are as follows: orientation 1/27; permutation 1/24; skip both (LL solved) 1/648. If superphasing is used, skip probability for both permutation as well as the entire last layer is twice as likely: permutation 1/12; skip both (LL solved) 1/324.

## Cons

• Phasing - Although not a great hindrance to move count, the phasing step adds an average of 2 7/12 moves. This may be offset by the increase in PLL skip probability and faster recognition.
• Superphasing - If edge permutation is executed every time, this adds an average of 6.5 moves (13 moves divided by 2 cases - parity or solved). This may be offset by the even greater PLL skip probability and recognition speed.